Remark 5.4 (Lemma REF )  
Using the Gauss–Lobatto nodes and their tensor-products to prove discrete trace inequalities is a known technique. The result of Lemma REF  however slightly differs from previous results from the literature and provides a sharper constant.
For instance, for \(p\ge 1\) , \(d=1\)  and \(k=0\) , [1]} leads to the constant \(\sqrt{2(p+1)^2}\)  and [2]} to the constant \(\sqrt{(p+1)(2p+1)}\) , which are both less sharp than \(C_1(0,p)\)  in (REF ). Notice also that (REF ) with \(d=1\)  leads to
\(\Vert  v^{\prime } \Vert _{\partial \tau } \le \sqrt{2p^2} h_\tau ^{-1/2} \Vert  v^{\prime } \Vert _{\tau }\)  which is again less sharp that (REF ) for \(p\ge 2\) .
Finally, we have the following multidimensional extension of Lemma REF  in a cuboid; the proof is omitted for brevity and follows arguments similar to those above.
Let \(\tau := [a_1, b_1] \times \ldots \times [a_d, b_d] \subset \mathbb {R}^d\) , with \(b_j > a_j\)  for all \(j\in \lbrace 1,\ldots ,d\rbrace \) , be a cuboid with boundary \(\partial \tau \) . Let \(p\ge 1\) . For any multi-index \(({k}) = (k_1, \ldots , k_d)\)  with \(k_j \in I_p\)  for all \(j\in \lbrace 1,\ldots ,d\rbrace \) , denoting the \({k}\) -th partial derivative of \(v\)  as \(v^{({k})}\) , the following holds:
\(\Vert  v^{({k})} \Vert _{\partial \tau } \le C_d({k},p,\tau ) \ \Vert  v^{({k})} \Vert _{\tau }, \quad \forall v \in \mathbb {Q}_p(\tau ),\) 
